Concept explainers
(a)
Find the required increased in speed at A
(a)
Answer to Problem 13.105P
The required increased in speed at point A
Explanation of Solution
Given information:
The altitude between the earth to point A
The altitude between the earth to point B
The radius of the earth (R) is
The acceleration due to gravity (g) is
Calculation:
Write the expression for the geocentric force acting on the spacecraft when it is orbiting around the earth
Here, G is the universal gravitational constant, M is the mass of the earth, and m is the mass of the space vehicle.
Write the expression for the force acting on the space vehicle on the surface of the earth due to gravity
Substitute
Substitute 9.81 m/s2 for g and
Write the expression for the centripetal force (F) acting on the spacecraft rotating around the earth at the given altitude as follows:
Here, v is the velocity of the spacecraft describing a circular path around the earth.
Write the expression for the geocentric force (F) acting on the spacecraft rotating at the given altitude around the earth as follows:
Equate equations (1) and (2).
Calculate the radius of the circular orbit described by the space vehicle around the earth at height
Substitute
Calculate the velocity of the space vehicle at point A
Substitute
Calculate the radius of the circular orbit described by the space vehicle around the earth at height
Substitute
Calculate the velocity of space vehicle at point B
Substitute
Use the principle of conservation of angular momentum states that, in the absence of external torque acting on the body, the angular momentum remains constant and no change of the momentum occurs during the entire process.
Write the expression for the principle of conservation of angular momentum as follows:
Here,
Substitute
Write the expression for the kinetic energy of the space vehicle at point A
Write the expression for the kinetic energy of the space vehicle at point B
Write the expression for the gravitational potential energy of the space vehicle at position A in the path AB
Write the expression for the gravitational potential energy of the space vehicle at position B in the path AB
Use the principle of conservation of energy states that sum of the kinetic and potential energy of a particle remains constant.
Calculate the speed of the space vehicle at positions A
Substitute
Substitute
Consider the equation (1).
Calculate the velocity of space shuttle at point A
Substitute
Calculate the increase in velocity at point A
Substitute
Calculate the increase in the velocity required at B
Substitute
Therefore, the required increased in speed at A
(b)
Find the total energy per unit mass
(b)
Answer to Problem 13.105P
The total energy per unit mass
Explanation of Solution
Given information:
The altitude between the earth to point A
The altitude between the earth to point B
The radius of the earth (R) is
The acceleration due to gravity (g) is
Calculation:
Calculate the total energy per unit mass of the vehicle to execute the transfer of space vehicle
Here, E is the total energy to execute the transfer.
Substitute
Therefore, the total energy per unit mass
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Chapter 13 Solutions
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